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little of help and suggestion from their teacher. What for example can be more unsatisfactory than the rule for Division of Fractions if blindly accepted and followed. “Invert the divisor and treat it as a multiplier.” This seems more like conjuring with numbers than performing a rational process. But suppose you first present the problem and determine to discover the ruie. You here find it needful to enlarge a little the conception of what Division means. “What is it" you ask “to divide a number?" It is

(1) To separate a number into equal parts;

(2) To find a number which multiplied by the divisor will make the dividend ;

(3) To find how many times, or parts of a time, the divisor is contained in the dividend.

It will have been shewn before, that this expression ‘the parts of a time' is necessary in dealing with fractions, and involves an extension of the meaning of the word divisor as ordinarily understood in dealing with integer numbers. You may then proceed to give four or five little problems graduated in difficulty, e.g.

(1) Divide 12 by j. What does this mean? To find how many times šis contained in 12. But } is contained three times in 1, so it must be contained 3 x 12 times in 12. Wherefore to divide by is the same as to multiply by 3.

(2) Divide 15 by 4. This means to find how many times are contained in 15. But I must be contained in it 15 * 4 or 60 times.

4X15 Sof must be contained in it one-third of 60 times or Where. fore to divide by { is the same as to multiply by 4. (3) Divide by 1. This means to divide by the fourth part of

5 3. Let us first divide by 3. Now divided by 3=;

7 x3 But since we were not to divide by three but by the fourth



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Demonstration of Fractional division.



part of 3, this result is too little, and must be set right by multiplying by 4. Hence

4 * 5

is the answer. Wherefore to divide by : is the same as to multiply by .

(4) To divide 4 by & is to find how often & is contained in. Let us bring them to a common denominator \=}, and = it. The question therefore is how often are it contained in aj? Just as often as 21 shillings are contained in 20 shillings; that is to say not once, but ji of a time, for this fraction represents the number of times that 20 contains 21. Wherefore 5+1=4 x š.

(5) To divide by d is to find a fraction which if multiplied by will make . That means that of this unknown fraction will make. But whenever A is fof B, B must be f of A. Hence the desired fraction must be of 4. But this is the same fraction which would have been produced by inverting the divisor and making it as a multiplier.

Wherefore to divide by any fraction is to multiply by its reciprocal, or

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I recommend that after each of these short exercises the numbers be altered, and the scholars required one by one to go through the demonstration orally. This will be found to serve exactly the same purpose as the proving of a theorem in geometry. It calls out the same mental qualities, demands concentration of thought, and careful arrangement of premisses and conclusion, and furnishes an effective though elementary lesson in logic and in pure mathematics.

The habit of registering the result of any such The use of process, or embodying any truth you have ascertained formulae. in the shape of a formula in which the letters of the alphabet are substituted for numbers is a very useful

The pupil makes a clear advance in abstract


thinking; if, for example, after shewing that equal additions to two numbers do not alter their difference, and illustrating this by such examples as

Because 12-7= 5, therefore (12 +S)-(7 + 8) = 5,

you help him also to see the truth of this :

If a-b=c, then (a + n) - (6 + n) = c.


Do not suppose that this is algebra. No one of the notions or processes proper to algebra is here involved. It is simply the statement of an arithmetical truth in its most abstract form. It lifts your pupil out of the region of particulars into the region of universal truths. It helps him to see that what is true of certain numbers and what he has actually verified in the case of those numbers is necessarily true of all numbers. So I recommend the practice of embodying each arithmetical truth

as you arrive at it in a general formula. Use of There is not a single process in Arithmetic out arithmeti

of which you may not get real intellectual training cal puzzles.

as well as practical usefulness, if you will only set this before you as one of the objects to be attained. The plea that it takes time, and hinders progress, is, in my opinion, wholly invalid. What do you mean by progress? It is surely not hastening to what are called advanced rules. It is rather such increased mastery of the fundamental principles of arithmetic as will enable the pupil to invent rules for himself. And this he will attain if you set him thinking about the meaning of every process which you require him to use.

Put before your class occasionally little facts about numbers, and ask them to find out the reasons for them. Here are two or three simple examples of what I mean:

Exercises in ingenuity and discovery.


(a) If the numbers in the following series progress by equal additions






3 5 7 9

13 · 15 17 19 · 21, why is it that each pair of numbers, e.g. the first and the last, the second and the last but one, the third and the last but two, &c. equals 22, a number equal to twice 11, the middle term ?

(6) If I take any number—say 732586, and any other composed of the same digits, say 257638, and subtract one from the other, thus :

732586 why is it that the digits of the remainder 257638

are sure to give me an exact number of 474948

nines 4+7+4+9+4+8 = 36 = 4 x 9? (c) If I take four numbers in proportion or representing two equal ratios, e.g. 6 : 24 :: 5

20, why is it that 6 times 20 must equal 24 x 5?

In this last case you will do well to make the scholar deduce the equality of the two products as a necessity from the fact that the four numbers are in proportion. He sees that 24 and 5 make a certain product, and because ex hypothesi 6 is as many times less than 24 as 20 is more than 5, therefore that the product of 6 and 20 must equal that of 5 and 24. And when this is seen to be necessarily true of all proportions, the ordinary rule for finding one of the factors when the other three are given will readily be supplied by the pupils themselves.

Proportion, however, though it is a very interesting Proporand valuable part of arithmetical science, and though its tion. principles furnish excellent opportunities for exercise in logical demonstration, is of less practical utility in the solution of problems than the text books seem to assume. The Rule of Three is a great stumbling-block to learners.

It comes much too early in the course, and learned empirically as it too often is, it is not readily capable of application to problems. Nearly all the questions usually set down under the head of 'Rule of Three' can be much better solved by simpler methods. Such a question as this for example :

“If 17 articles cost £23. 1os., what will 50 such articles cost?"

ought not to be stated and worked as Proportion; but
by the method of reduction to unity, thus :-
One article must cost £23. 105. – 17. Therefore

£23. 105. x 50
50 articles must cost

17 Thus the true place for the theory of proportion is after fractions, vulgar and decimal, have been well

understood and seen in varied applications. Extraction My last illustration shall be taken from an advanced of roots.

rule, that for the Extraction of the Square Root. I will, as before, take an easy sum, and the directions for solving it, as given in the ordinary books.

Find the square root of 676, or the number which, multiplied by itself, will give 676.

RULE—“ Point off the alternate

numbers from the unit, and thus divide 676 ( 2 the numbers into periods. 4

“Find the nearest square root of 46 ) 276 ( 6

the first period, and subtract its square. 276

“(The nearest square root of 6 is 2;

set down 2, and take twice 2 from 6.) "Set down the remainder, and bring down the next period.

“(Set down 2 and bring down 76.)

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